# Using Quantum Numbers in Particle Physics

Wow, that title seems rather intimidating; let's see if we can calm your fears. There will be no complex mathematics, and no advanced mathematical symbols, in fact, you will not even need to look at a calculator to understand this. However, we need to make a run through of some basic terms.

### Some Basic Terms

• Matter - This is the solid stuff we are made up of.

• Energy - This is the other stuff that makes things happen, it is associated with movement, heating and light.

• Decay - This is when one particle turns into one or more, generally smaller, particles.

• Fundamental Particle - These are particles that cannot be broken up into anything smaller. We used to think atoms were fundamental, then we thought protons were, now we think quarks are. Who knows in the next hundred years what we will think are fundamental.

• Fundamental Force - There are four forces that govern interactions between matter. We shall ignore gravity for this entry as it is very weak and doesn't do much to individual particles. That leaves us with the strong force ... which is strong, the weak force, which is weak1, and the electromagnetic force which is in the middle and works on any particle with charge.

• Antimatter - Every particle has a corresponding antiparticle which is exactly the same except all its quantum numbers are the opposite sign (so a positive charge will become negative). Antimatter and matter tend to blow each other up if they get too close.

• Charge - In this context, it isn't what the Light Brigade did at the Battle of Balaclava. It is, in fact, a measure of how strongly something interacts with electromagnetic forces. A particle with charge is called a charged particle. For example, protons are positively charged and electrons are negatively charged.

• Spin - Particles have spin. Well, they don't really spin, but given that the human brain finds it very hard to comprehend the actual dynamics of the sub-atomic universe, we just decided to use a name that was very easy to visualise to make it easier to explain.

• Colour - Some sub-atomic particles have colour. Well, they don't really, because the size of the waves of coloured light are much, much larger than atoms, let alone sub-atomic particles, so they aren't affected by them. We just use colour, like we use spin, as an analogy to make it easier to visualise and talk about.

• Lepton - these are fundamental particles that are lucky in they get to ignore the strong force.

• Electron - these are negatively charged leptons. We normally find them buzzing around atoms or carrying current in electric circuits. All electrons are alike, so alike in fact that Richard Feynman pointed out that it was possible that there was only one electron in the entire universe and it just happens to travel through time2.

• Muon - these are fat electrons, they do exactly the same thing as electrons, but are larger.

• Tau - these are obese electrons, they pretty much do the same as muons and electrons but decay very quickly.

• Neutrino - these are leptons with no charge. Since they have little or no mass, and no charge, they get to ignore the strong force, gravity and the electromagnetic force. There are three types of neutrino, associated with the electron, the muon and the tau. Particle Physicists assure us there is actually some difference between the three of them.

• Antimatter - Just to remind you that the electron, the muon and the tau, along with all their neutrinos have their own antiparticles.

• Quark - These have nothing to do with being bar tenders of a space station, instead they are fundamental particles used as building blocks for other particles. Quarks don't normally appear on their own: it's nice to have friends. There are six types of quark. The smallest are the up and down. Then there are the strange and charm, then we have truth and beauty3. It was decided that despite truth and beauty being the goal of all good scientists, these were silly names for particles and they got renamed top and bottom. Which aren't silly at all. Top quarks decay very quickly, in fact the only quarks we normally encounter are up and down.

• Baryon - These are particles made out of three quarks. They include the proton and neutron.

• Meson - These are particles made up from a quark and an antiquark. Most mesons last less than a millionth of a second which is a very short time in the real world, but ages in the weird world of particle physics.

• Antimatter - In case we'd forgotten, all six quarks plus all the baryons and mesons have their own antiparticles.

### So what are Quantum Numbers?

These are numbers that describe a property of a particle, we have met one already: charge. Quantum means that they have to have certain set values, such as +1, 0 -1 or perhaps a fraction. They cannot vary over a continuous range of numbers.

Let's look at charge. Neutrons and neutrinos have a charge of 0. Electrons, muons and taus have a charge of -1 and protons are +1. Remember that antimatter has opposite quantum numbers so an antielectron4 is +1 and an antiproton is -1. But what about quarks? Up, top and charm have charges of +2/3. Down, bottom and strange are -1/3.

We will introduce two more now, these are quite simple too. They are the lepton number and the baryon number. If a particle is a lepton, then its lepton number is +1, if it is not, then it is 0. If a particle is a baryon, then it has a baryon number of +1, it not, it is 0. Well, not quite, because three quarks make up a baryon, so a quark has a baryon number of +1/3.

While we are here, let's stick all the cards on the table and throw in some more numbers. We will start with strangeness. A strange quark has a strangeness of -1. All other quarks have a strangeness of 0. Likewise, a bottom quark has a bottomness of -1. The two remaining quarks with positive charge have a charm of +1 (charm quark) and a topness of +1 (top quark5). The antiparticles will have the opposite quantum number, so an antistrange quark has strangeness of +1.

### So what do Quantum Numbers do?

They help us work out the structure of particles and what happens in interactions.

#### So what about Making up Particles?

Okay, let's take the example of a proton. We know that a proton has a charge of +1, a baryon number of +1 and no charm, strangeness, topness or bottomness. We also know it is a baryon, so will be made up of three quarks. Since it has no charm, strangeness, topness or bottomness, we must conclude that it can only consist of up and downs. Now we look at the charge. The only way to made +1 from the charges on the quarks is to have two sets of +2/3 (making +4/3) and then -1/3, making 3/3, which is the same as +1. In terms of quarks this will be up, up, down.

So an antiproton will have a charge of -1 and a baryon number of -1. Using the same principles we find an antiproton is made up of antiup, antiup and antidown. As expected with antimatter, this is the exact opposite of a proton.

So what about a strange particle? We can look at a positive kaon (K+). A kaon is a meson with a strangeness of +1, and because it is K+ it has a charge of +1. Mesons have a baryon number of 0 which means we have to have a quark and an antiquark. A strange quark has a strangeness of -1, so we need an antistrange quark. An antistrange quark has a charge of +1/3, so we need a normal quark with charge of +2/3 to complete the meson, this means that we have to use an up quark.

One method of radioactive decay is beta decay. This is when a neutron in an atom decays to a proton and it fires out a beta particle. A beta particle is actually a very fast moving electron. In all interactions6 the charge, baryon and lepton numbers are conserved. This means that their sums remain the same after decay as they were before. So how does this work in this example?

1. Charge - We start off with a neutron with a charge of 0, we end up with a proton (+1) and an electron (-1) which totals up to 0. Charge is conserved.

2. Baryon Number – We start off with a neutron (+1) and end up with a proton (+1) and an electron (0). Baryon number is conserved.

3. Lepton Number – We start off with a neutron (0) and end up with a proton (0) and an electron (+1). The Lepton number isn't conserved here yet so we need another particle. We know it fits in with the other numbers, it has to have 0 charge, so we are looking for a neutral particle with a lepton number of -1. Since it has a lepton number of -1, we know it is an antilepton. We are looking therefore for an antineutrino. To be more specific, because the interaction involved an electron, the final particle is an electron antineutrino.

We now know that a neutron decays into a proton, a beta particle (electron) and an electron antineutrino.

Let's try and get a bit more exotic and see what happens when a positive pion decays.

The pion has a charge of +1, and it has baryon, lepton and strangeness of 0. It has to decay so that the resulting particles add up to charge +1, with 0 total baryon and lepton. It turns out that pions decay into leptons and that since pions are a bit big, they actually turn into muons. Because we have a charge of +1 on the pion, we need a muon with charge of +1, which means we produce an antimuon. The antimuon has a lepton number of -1, so we need a normal lepton to balance it, and this lepton needs a charge of 0 to balance the charge. This means we produce a neutrino, specifically a muon neutrino.

So a positive pion decays into an antimuon and a muon neutrino.

### Colouring it in

So we know that we can use quantum numbers to figure out how a hadron is structured. We also know that we can use conservation of charge, lepton number and baryon number to figure out interactions like radioactive decay, but can we go further? Yes... we can leave numbers far behind and return to colours.

Just as particles have electric charge, quarks have colour charge. This charge determines how they interact with the strong nuclear force in the theory of Quantum Chromodynamics (QCD). Quarks can be red, blue or green. Of course, this is rubbish. As stated earlier, quarks are too small to actually have a real colour. In the quantum world, colour is just a useful analogy7. What we really mean to say is that quarks can have a red colour charge, a green colour charge or a blue colour charge.

Since we can't use quarks for picking paint samples, what is the point of these colours? Well it turns out that in our analogy hadrons have to be colourless. As any physicist or computer graphics expert will tell you, if you mix red, blue and green light, you get white which is colourless. This means that every baryon must have one red quark, one green quark and a blue quark. What about antibaryons? Well, it turns out that antiquarks have antired (cyan), antiblue (yellow) and antigreen (magenta). If you ask artists what happen when you mix inks of these colours together, they will tell you that you get black, which is colourless. This means that in an antibaryon, you will have one antiblue, one antired and one antigreen quark.

So what about mesons? They only have two quarks, so what happens here? Well here, in order to remain colourless, the antiparticle must have the corresponding anticolour to the particle. This means that if you have a blue quark, then the antiquark must be antiblue.

It is important to say that the colour of quarks do not have any relationship to their flavour (up, down, strange, charm etc).

### To sum it all up

• All particles have quantum numbers, these include charge, spin, baryon number, lepton number and strangeness.

• All particles have antiparticles, the antiparticle having equal but opposite quantum numbers.

• To find what fundamental particles are used to build a larger particle, we know that the sum of each of the quantum numbers of the individual fundamental particles adds up to the corresponding quantum number of the big particle. For example: the charge of the quarks in a proton adds up to charge of the proton.

• In particle interactions, the lepton number, baryon number and charge are conserved.

1Sometimes these are called the strong nuclear force and weak nuclear force, because they mostly affect atomic nuclei.2Don't think about that too much, as it will lead to headaches after a while.3We can blame Murray Gell-Mann for a lot of the stupid names. He took the name for quark from a line from a James Joyce book and tried to blend Buddhist philosophy into the names he used for his particle theories. Others carried on his tradition.4Positron.5Although never produced for the BBC, one might assume that Richard Hammond would be of suitable size to present this program.6Well, not exactly all, but all worth mentioning.7Because there are three different charges, physicists couldn't use + and -, as with electric charge, so they chose something there are three of: primary colours.

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